
where \(\tilde{X}_S\)  is a random variable based on the distribution of \(X_S\) .
There are two versions of PFI, the marginal PFI and the conditional PFI, which have different strategies to replace \(X_S\)  and also different interpretations.
The marginal PFI can be interpreted as the importance of the feature, ignoring dependencies with other features and also ignoring that the data used may differ greatly from the original joint distribution \({P}_X\)  (extrapolation).
For the marginal PFI we take the expected value over the distribution \({P}_{X_S} \cdot {P}_{X_CY}\) , which means that \(\tilde{X}_S\)  follows the marginal distribution of \(X_S\)  and is independent of \(X_C\)  and \(Y\)  (\(\tilde{X}_S \rotatebox {90}{\XMLaddatt {origin}{c}\models }X_C,Y\) ).
This means that the marginal PFI breaks the association between the feature(s) \(X_S\)  and the target \(Y\) , but also between \(X_S\)  and all other features \(X_C\) .
For the conditional PFI (cPFI) [1]}, [2]}, [3]}, [4]}, the expectation is taken over the distribution \({P}_{X_S|X_C} \cdot {P}_{X_CY}\) , so that \(\tilde{X}_S\)  follows the conditional distribution of \(X_S\)  given \(X_C\)  but is still independent of \(Y\) .
The interpretation of the conditional PFI of a feature is therefore also conditional on all features that are correlated with the feature of interest.
Conditional PFI may be interpreted as the additional importance of a feature given that we already know the other feature values.
